On the skew representations of the quantum affine algebra

An explicit realization of the skew representations of the quantum affine algebra U _q (gl _n ) is given. It is used to identify these representations in a simple way by calculating their highest weight, Drinfeld polynomials and the Gelfand-Tsetlin character (orq-character).     

A remark on the Gelfand-Tsetlin patterns for symplectic groups

In this paper, we study an approach by Gelfand-Tsetlin to the representation of symplectic groups.     

The representations and coupling coefficients of su(n); application to su(4)

Analytical expressions for the matrices and an explicit algorithm for computing Clebsch-Gordan coupling coefficients are given forsu(4) in au(3)-coupled basis as an example of the construction for anysu(n) in au(n−1) basis. The results areinduced from the known results foru(3) by means of the vector-coherent-state (VCS) theory of induced representations. The important recent result that makes this possible is the discovery that a complete set of shift tensors for the finitedimensional representations of reductive Lie algebras can be induced, by VCS methods, from those of suitably defined subalgebras.     

Spinor-vector duality in heterotic string vacua

Classification of the N=1 space–time supersymmetric fermionic Z₂×Z₂ heterotic-string vacua with symmetric internal shifts, revealed a novel spinor-vector duality symmetry over the entire space of vacua, where the S_t↔V duality interchanges the spinor plus anti-spinor representations with vector representations. In this paper we demonstrate that the spinor-vector duality exists also in fermionic Z₂ heterotic string models, which preserve N=2 space–time supersymmetry. In this case the interchange is between spinorial and vectorial representations of the unbroken SO(12) GUT symmetry. We provide a general algebraic proof for the existence of the S_t↔V duality map. We present a novel basis to generate the free fermionic models in which the ten-dimensional gauge degrees of freedom are grouped into four groups of four, each generating an SO(8) modular block. In the new basis the GUT symmetries are produced by generators arising from the trivial and non-trivial sectors, and due to the triality property of the SO(8) representations. Thus, while in the new basis the appearance of GUT symmetries is more cumbersome, it may be more instrumental in revealing the duality symmetries that underly the string vacua.     

A unified treatment of the groups SO(4) and SO(3,1)

The irreducible representations of the group SO(4) in which the SO(3) subgroup is reduced are studied by an explicit construction of the operators and the basis in the spinor representation. The basis function which is formally identical with that for the coupling of two angular momentaj ₁ andj ₂ is expressible in terms of a hypergeometric function and strongly resembles the one for the irreducible representations of the groups SO(3,1). For the Lorentz group, the bases for the unitary representations which require unphysical values ofj ₁ andj ₂ are found to be analytic continuation of those for SO(4). The realization of the unitary irreducible representations of the group SO(4) in the Hilbert space of these functions leads, for appropriate unphysical values ofj ₁,j ₂, to the Gelfand-Naimark formula for the principal and complementary series of the representations of SO(3;1). The matrix elements for finite transformations of SO(4) and SO(3,1) can be evaluated, in this approach, in a unified manner by using standard properties of the hypergeometric function. These turn out to be a finite sum of₃ F ₂-functions which, as expected, are polynomials for SO(4) and infinite series for SO(3,1). A number of special matrix elements are calculated from the general formula and these agree with the results obtained previously.The authors are deeply indebted to Professor S.Dutta Majumdar fo many important suggestions and clarifications.     

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Explicit expressions for the generators of the quantum superalgebraU _q [gl(n/m)] acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq-number identities.     

The jordanian deformation of Su(2) and clebsch-gordan coefficients

Representation theory for the Jordanian quantum algebraU _h (sl(2)) is developed using a nonlinear relation between its generators and those of sl(2). Closed form expressions are given for the action of the generators ofU _h (sl(2)) on the basis vectors of finite dimensional irreducible representations. In the tensor product of two such representations, a new basis is constructed on which the generators ofU _h (sl(2)) have a simple action. Using this basis, a general formula is obtained for the Clebsch-Gordan coefficients ofU _h (sl(2)). Some remarkable properties of these Clebsch-Gordan coefficients are derived. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Particle Weights and Their Disintegration II

The first article in this series presented a thorough discussion of particle weights and their characteristic properties. In this part a disintegration theory for particle weights is developed which yields pure components linked to irreducible representations and exhibiting features of improper energy-momentum eigenstates. This spatial disintegration relies on the separability of the Hilbert space as well as of the C*-algebra. Neither is present in the GNS-representation of a generic particle weight so that we use a restricted version of this concept on the basis of separable constructs. This procedure does not entail any loss of essential information insofar as under physically reasonable assumptions on the structure of phase space the resulting representations of the separable algebra are locally normal and can thus be continuously extended to the original quasi-local C*-algebra.     

Representation for matrix elements of compact Lie groups

A method for evaluation of matrix elements of unitary irreducible representations of compact Lie groups is formulated. matrix elements of some representations of the group SO(p+q) in the SO(p) SO(q) basis and of the group U(p) U(p) in the U(p) basis (diagonal imbedding) are given.     

Discrete series of unitary irreducible representations of the Uq(u(3, 1)) and Uq(u(n, 1)) noncompact quantum algebras

The structure of all discrete series of unitary irreducible representations of the U _q (u(3, 1)) and U _q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) _q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U _q (u(n, 1)) ⊇ U _q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.     

Derivation of O(3) Electrodynamics from the Einstein–Sachs Theory of General Relativity

General relativity is reduced to O(3) electrodynamics by consideration of the irreducible representations of the Einstein group and through a particular choice of basis. The photon is shown always to possess a scalar curvature R, and so the origin of quantization is found in general relativity.     

Group-Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup SU(2, 2 /N) induced from irreducible finite-dimensional Lorentz and SU(N) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Examples of higher order invariant differential operators are given.     

Indefinite harmonic forms and gauge theory

Indecomposable representations have been extensively used in the construction of conformal and de Sitter gauge theories. It is thus noteworthy that certain unitary highest weight representations have been given a geometric realization as the unitary quotient of an indecomposable representation using indefinite harmonic forms [RSW]. We apply this construction toSU (2,2) and the de Sitter group. The relation is established between these representations and the massless, positive energy representations ofSU (2,2) obtained in the physics literature. We investigate the extent to which this construction allows twistors to be viewed as a gauge theory ofSU (2,2). For the de Sitter group, on which the gauge theory of singletons is based, we find that this construction is not directly applicable.     

An extended Galilei group and its application

The Galilei group is combined with two one-dimensional groups, to form a twelve-dimensional extended Galilei group. Irreducible representations of this group depend upon two indicesm, s that can, respectively, be interpreted as the mass and spin of a non-relativistic particle. It turns out that the true irreducible representations of the ordinary Galilei group correspond tom=0, and this explains why these representations have no physical interpretation.     

Matrix elements for the representations of SO(n) and SO0(n, 1)

Matrix elements of the unitary irreducible representations of the group SO(n) of class higher then 1 (with respect to SO(n−1)) in Gel'fand-Zetlin basis are obtained in explicit form. They are represented as polynomials in cosθ and sinθ of the order equal to the first coordinate of the highest weight. Making use of them the representation matrix elements for the group SO₀(n, 1) in SO(n) basis are calculated.     

High efficiency redundant binary number representations for parallel arithmetic on optical computers

A family of redundant binary number representations, obtained by generalization of the RB (redundant binary) number representation, is introduced. All these number representations are suitable for optical computing and have properties similar to the RB representation. In particular, the p-RB (packed redundant binary) number representation introduced in this work has efficiency greater than both RB and MSD (modified signed digit) representations. With p-RB numbers the algebraic sum is always permitted in constant time for any efficiency value. p-RB representations also fit in a natural way the 2's complement binary number system. Symbolic substitution truth tables for the algebraic sum and several examples of computation are also given.     

Finite-dimensional representations of the euclidean group in the plane

In this article two theorems are given which permit, together with the concept of a representation vector diagram, to classify all (linear) finite-dimensional representations of the algebra and group E ₂ which are induced by a master representation on the place of the universal enveloping algebra of the algebra E ₂. Apart from a classification of the finite-dimensional representations, the two theorems make it possible to obtain the matrix elements of these representations for both, algebra and group, in explicit form. The material contained in this letter forms part of an analysis of indecomposable (finite- and infinite-dimensional) representations of the algebra and group E ₂ which is contained in Reference [1]. No proofs will be given in this letter. We refer instead to [1].     

Modeling of IR Spectra of Compounds on the Basis of Analogy between Fragmental Compositions of Structures

The possibility of using representations of the structures of compounds in the form of nonisomorphic k-vertex connective fragments for modeling IR spectra of compounds with a given structure on the basis of the spectra of their close structural analogs selected from a database is shown. Statistical justification of the approach and examples of modeled spectra are given.     

∗-Representations of the Quantum Algebra U q(sl(3))

Abstract

Studied in this paper are real forms of the quantum algebra U _q(sl(3)). Integrable operator representations of ?-algebras are defined. Irreducible representations are classified up to a unitary equivalence.